Markov Chains & Long-Run Behavior · Superchallenge problem

The Petersen graph has $10$ vertices: the $2$-element subsets of $\{1, 2, 3, 4, 5\}$, with two subsets joined by an edge when they are disjoint. Each vertex has degree $3$. A walker moves each step to a uniformly random neighbour. Starting at $\{1, 2\}$, what is the expected number of steps until it first reaches $\{1, 3\}$?